Question

The function $f\left(x\right)=\frac{x{e}^{1/x}}{1+e^{1/x}}+\sin \left(1/x\right),f\left(0\right)=0$ at x=0 is

• A. continuous
• B. discontinuous
• C. cannot be determined
• D. none

Answer 1

The correct option is B discontinuous

Given,

$f\left(x\right)=\frac{x{e}^{\frac{1}{x}}}{1+e^{\frac{1}{x}}}+sin\frac{1}{x}$

$f\left(0\right)=0$ at $x=0$

$Takethedenominator\left(s\right)of\frac{x{e}^{\frac{1}{x}}}{1+e^{\frac{1}{x}}}+\sin \left(\frac{1}{x}\right)andcomparetozero$

$x=0$

$x<0orx>0$

$\left[\begin{array}{cc}Solution:& x<0orx>0\\ IntervalNotation:& (-\infty ,0)\cup (0,\infty )\end{array}\right]$

Therefore $f\left(x\right)$ is discontinuous.